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Theorem moabs 2627
 Description: Absorption of existence condition by uniqueness. (Contributed by NM, 4-Nov-2002.) Shorten proof and avoid df-eu 2655. (Revised by BJ, 14-Oct-2022.)
Assertion
Ref Expression
moabs (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))

Proof of Theorem moabs
StepHypRef Expression
1 ax-1 6 . 2 (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃*𝑥𝜑))
2 nexmo 2625 . . 3 (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
3 id 22 . . 3 (∃*𝑥𝜑 → ∃*𝑥𝜑)
42, 3ja 189 . 2 ((∃𝑥𝜑 → ∃*𝑥𝜑) → ∃*𝑥𝜑)
51, 4impbii 212 1 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∃wex 1781  ∃*wmo 2622 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971 This theorem depends on definitions:  df-bi 210  df-ex 1782  df-mo 2624 This theorem is referenced by:  mo3  2649  mo4  2651  moeu  2669  dffun7  6363  wl-mo3t  34877
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